Lecture 3: Business Values Discounted Cash flow, Section 1.3 © 2004, Lutz Kruschwitz and Andreas Löffler

1.3.1 Trade and payment dates The time structure of the model: 1.Buy in t-1  pay price. 2.Hold until t  receive dividend. 3.Sell (and buy again) in t  receive (and pay) price. Notice that the investor receives dividend shortly before t (the selling date).

1.3.1 A riskless world What should happen if the future where certain? In a certain world that is free of arbitrage we must have Otherwise, if for example  take loan, buy share in t, wait until t+1, get dividend and sell share  get infinitely rich without any cost.

1.3.1 A risky world In a risky world we have What is absence of arbitrage now? The account of the largest takeover in Wall Street history...

1.3.1 Three roads lead to Rome Certainty equivalent Risk-neutral probabilities Risk premium

1.3.1 Three roads lead to Rome Certainty equivalent Risk premium Risk-neutral probabilities

1.3.1 Three roads lead to Rome Certainty equivalent Risk premium Risk-neutral probabilities

1.3.1 Roads to Rome: example We are going to illustrate the three roads by using the following example: And r f = Because the cash flows have expectation The value V 0 will be less than

1.3.1 Certainty Equivalent If we assume a utility function Then the «certainty equivalent» (CEQ) and hence the price of the asset is given by Daniel Bernoulli, founder of utility theory

1.3.1 Risk Premium Valuation with a «risk premium» uses another idea: we modify the denominator Using the numbers of the example we get

1.3.1 Risk-neutral Probabilites With «risk-neutral probabilities» the probabilities p u =0.5 and p d =0.5 are modified. The question is: find q u and q d such that Holds. Answer: We turn to this approach in more detail!

1.3.2 Cost of capital Cost of capital is definitely a key concept. But: how is it precisely defined?

1.3.2 Cost of capital – alternative definitions There are several alternative definitions of cost of capital in a multi-period context: cost of capital = Def «Yields» cost of capital = Def «Discount rates» cost of capital = Def «Expected returns» cost of capital = Def «Opportunity costs» Are all definitions identical? We will show later: not necessarily!

1.3.2 Cost of capital is an expected return Which definition is useful for our purpose? The following definition turns out to be appropriate Definition 1.1 (cost of capital): The cost of capital of a firm is the conditional expected return

1.3.2 Shortcoming of definition This appropriate definition has a disadvantage: k t could be uncertain. And you cannot discount with uncertain cost of capital! Another (big) assumption is necessary: from now on this cost of capital should be certain.

1.3.2 Alternative definitions? Other attempts to simplify definition 1.1 fail to produce reasonable results: aim is to have as well as for t=1 with the same cost of capital in the denominator!

1.3.3 Market value Theorem 1.1 (market value): When cost of capital is deterministic, then Notice that the lefthand side and the righthand side as well can be uncertain for t>0.

1.3.3 Proof of Theorem 1.1 Start with a reformulation of Definition 1.1, Now use the similar relation for and plug in,

1.3.3 Proof (continued) Costs of capital are deterministic, use rule 2 (linearity) Rule 4 (iterated expectation) gives

1.3.3 Proof (continued) Continue until T to get Last term vanishes by transversality. QED

1.3.4 Fundamental Theorem Let us turn back to risk-neutral probability Q: does Q always exist? This is not a trivial question: the numbers q u and q d must be between 0 and 1! For example,would not be considered as «probabilities». Theorem 1.2 (Fundamental Theorem): If the markets are free of arbitrage, there is a probability Q such that for all claims

1.3.4 Fundamental theorem What about a proof? Forget it. 1 How to get Q for valuation of firms? No idea. So why is this helpful? We will see (much) later. Is there at least an interpretation of Q? Yes! 1 If you cannot resist: see further literature

1.3.4 Risk-neutrality of Q Why do we call Q a risk-neutral probability? Look at the following:

1.3.4 Intuition of Q The last equation simply says: if we change our probabilites to Q, any security has expected return r f. Or: the world is risk-neutral under Q.

1.3.4 Uniqueness of Q Is Q unique? Or can the value of the company depend on Q? If the cash flows of the firm can be duplicated by traded assets (market is complete) then any Q will lead to the same value. Proof: Forget this as well...

1.3.4 Two assumptions From now on two assumptions will always hold: Assumption 1.1: The markets are free of arbitrage. Assumption 1.2: The cash flows of the firm can be duplicated by traded assets.  The risk-neutral probability Q exists and is (in some sense) unique.

Summary Costs of capital are conditional expected returns. Costs of capital must be deterministic. If markets are arbitrage free a «risk-neutral probability measure» Q exists. When using this probability Q the world is risk-neutral.